Average Error: 0.2 → 0.0
Time: 50.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r73038777 = a;
        double r73038778 = r73038777 * r73038777;
        double r73038779 = b;
        double r73038780 = r73038779 * r73038779;
        double r73038781 = r73038778 + r73038780;
        double r73038782 = 2.0;
        double r73038783 = pow(r73038781, r73038782);
        double r73038784 = 4.0;
        double r73038785 = r73038784 * r73038780;
        double r73038786 = r73038783 + r73038785;
        double r73038787 = 1.0;
        double r73038788 = r73038786 - r73038787;
        return r73038788;
}
↓
double f(double a, double b) {
        double r73038789 = 4.0;
        double r73038790 = b;
        double r73038791 = r73038790 * r73038790;
        double r73038792 = r73038789 * r73038791;
        double r73038793 = a;
        double r73038794 = r73038793 * r73038793;
        double r73038795 = r73038791 + r73038794;
        double r73038796 = sqrt(r73038795);
        double r73038797 = pow(r73038796, r73038789);
        double r73038798 = r73038792 + r73038797;
        double r73038799 = 1.0;
        double r73038800 = r73038798 - r73038799;
        return r73038800;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot 4 + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right) - 1}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}\right) - 1\]

  5. Applied associate-*r*0.1

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}}\right) - 1\]
  6. Using strategy rm

  7. Applied pow10.1

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}}\right) - 1\]

  8. Applied add-sqr-sqrt0.1

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}\right) - 1\]

  9. Applied pow30.1

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}\right) - 1\]

  10. Applied pow-prod-up0.0

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\left(3 + 1\right)}}\right) - 1\]

  11. Simplified0.0

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\color{blue}{4}}\right) - 1\]

  12. Final simplification0.0

    \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))